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1. Supersingular elliptic curves over ℤ _𝑝 -extensions

   https://doi.org/10.1515/crelle-2023-0029  

   Çiperiani, Mirela (June 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let $\mathrm{E}/\mathbb{Q}$\mathrm{E}/\mathbb{Q}be an elliptic curve and 𝑝 a prime of supersingular reduction for $\mathrm{E}$\mathrm{E}.Consider a quadratic extension $L/{\mathbb{Q}}_p$L/\mathbb{Q}_{p}and the corresponding anticyclotomic ${\mathbb{Z}}_p$\mathbb{Z}_{p}-extension $L_{\mathrm{\infty }}/L$L_{\infty}/L.We analyze the structure of the points $\mathrm{E}\left(L_{\mathrm{\infty }}\right)$\mathrm{E}(L_{\infty})and describe two global implications of our results. 

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2. A note on rank one quadratic twists of elliptic curves and the non-degeneracy of 𝑝-adic regulators at Eisenstein primes

   https://doi.org/10.1090/bproc/144  

   Burungale, Ashay; Skinner, Christopher (February 2023, Proceedings of the American Mathematical Society, Series B)

   We show that for certain non-CM elliptic curves E / Q E_{/\mathbb {Q}} such that 3 3 is an Eisenstein prime of good reduction, a positive proportion of the quadratic twists E ψ E_{\psi } of E E have Mordell–Weil rank one and the 3 3 -adic height pairing on E ψ ( Q ) E_{\psi }(\mathbb {Q}) is non-degenerate. We also show similar but weaker results for other Eisenstein primes. The method of proof also yields examples of middle codimensional algebraic cycles over number fields of arbitrarily large dimension (generalized Heegner cycles) that have non-zero p p -adic height. It is not known – though expected – that the archimedian height of these higher-codimensional cycles is non-zero. 

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3. Inclusive $$\text {J}/\psi $$ production at midrapidity in pp collisions at $$\sqrt{s} = 13$$ TeV

   https://doi.org/10.1140/epjc/s10052-021-09873-4  

   Acharya, S.; Adamová, D.; Adler, A.; Aglieri Rinella, G.; Agnello, M.; Agrawal, N.; Ahammed, Z.; Ahmad, S.; Ahn, S. U.; Ahuja, I.; et al (December 2021, The European Physical Journal C)

   Abstract We report on the inclusive $$\text {J}/\psi $$ J / ψ production cross section measured at the CERN Large Hadron Collider in proton–proton collisions at a center-of-mass energy $$\sqrt{s}~=~13$$ s = 13  TeV. The $$\text {J}/\psi $$ J / ψ mesons are reconstructed in the $$\text {e}^{+}\text {e}^{-}$$ e + e - decay channel and the measurements are performed at midrapidity ( $$|y|<0.9$$ | y | < 0.9 ) in the transverse-momentum interval $$0<40$$ 0 < p T < 40  GeV/ $$c$$ c , using a minimum-bias data sample corresponding to an integrated luminosity $$L_{\text {int}} = 32.2~\text {nb}^{-1}$$ L int = 32.2 nb - 1 and an Electromagnetic Calorimeter triggered data sample with $$L_{\text {int}} = 8.3~\mathrm {pb}^{-1}$$ L int = 8.3 pb - 1 . The $$p_{\mathrm{T}}$$ p T -integrated $$\text {J}/\psi $$ J / ψ production cross section at midrapidity, computed using the minimum-bias data sample, is $$\text {d}\sigma /\text {d}y|_{y=0} = 8.97\pm 0.24~(\text {stat})\pm 0.48~(\text {syst})\pm 0.15~(\text {lumi})~\mu \text {b}$$ d σ / d y | y = 0 = 8.97 ± 0.24 ( stat ) ± 0.48 ( syst ) ± 0.15 ( lumi ) μ b . An approximate logarithmic dependence with the collision energy is suggested by these results and available world data, in agreement with model predictions. The integrated and $$p_{\mathrm{T}}$$ p T -differential measurements are compared with measurements in pp collisions at lower energies and with several recent phenomenological calculations based on the non-relativistic QCD and Color Evaporation models. 

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4. $$p^\infty $$-Selmer groups and rational points on CM elliptic curves

   https://doi.org/10.1007/s40316-022-00203-y  

   Burungale, Ashay; Castella, Francesc; Skinner, Christopher; Tian, Ye (July 2022, Annales mathématiques du Québec)

   Résumé Soit $$E/{\mathbb {Q}}$$ E / Q une courbe elliptique à multiplication complexe et p un nombre premier de bonne réduction ordinaire pour E . Nous montrons que si $${\mathrm{corank}}_{{\mathbb {Z}}_p}{\mathrm{Sel}}_{p^\infty }(E/{\mathbb {Q}})=1$$ corank Z p Sel p ∞ ( E / Q ) = 1 , alors E a un point d’ordre infini. Le point de non-torsion provient d’un point de Heegner, et donc $${{{\mathrm{ord}}}}_{s=1}L(E,s)=1$$ ord s = 1 L ( E , s ) = 1 , ce qui donne une réciproque à un théorème de Gross–Zagier, Kolyvagin, et Rubin dans l’esprit de [49, 54]. Pour $$p>3$$ p > 3 , cela donne une nouvelle preuve du résultat principal de [12], que notre approche étend à tous les nombres premiers. L’approche se généralise aux courbes elliptiques à multiplication complexe sur les corps totalement réels [4]. 

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5. The geometric distribution of Selmer groups of elliptic curves over function fields

   https://doi.org/10.1007/s00208-022-02429-1  

   Feng, Tony; Landesman, Aaron; Rains, Eric M. (September 2022, Mathematische Annalen)

   Abstract Fix a positive integernand a finite field$${\mathbb {F}}_q$$ $F_q$. We study the joint distribution of the rank$${{\,\mathrm{rk}\,}}(E)$$ $\mathrm{rk}\left(E\right)$, then-Selmer group$$\text {Sel}_n(E)$$ ${\text{Sel}}_n\left(E\right)$, and then-torsion in the Tate–Shafarevich group Equation missing<#comment/>asEvaries over elliptic curves of fixed height$$d \ge 2$$ $d\ge 2$over$${\mathbb {F}}_q(t)$$ $F_q\left(t\right)$. We compute this joint distribution in the largeqlimit. We also show that the “largeq, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains. 

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